From BRYAN@wvnvm.wvnet.edu Thu May 11 03:58:11 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04002; Thu, 11 May 95 03:58:11 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 9222; Wed, 10 May 95 22:38:33 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 9300; Wed, 10 May 1995 22:38:33 -0400 Message-Id: Date: Wed, 10 May 1995 22:38:32 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: more on the slice group In-Reply-To: Message of 05/09/95 at 12:11:02 from hoey@AIC.NRL.Navy.Mil On 05/09/95 at 12:11:02 hoey@AIC.NRL.Navy.Mil said: >No, the 4-spot pattern is also a local maximum at 12 qtw, although its >symmetry group is of order 16. Jim Saxe and I reported this on 22 >March 1981, in "No short relations and a new local maximum". Argh! After Dan and Mike pointed this out, I did remember having seen it in the archives. Worse still, Dan pointed it out again on 3 August 1992. But since it has come up, let's take a brief look at the 22 March 1981 note. > With five-qtw searches, it became possible to check another >conjecture, using an approach that Jim suggested. The four-spot >pattern > > U U U > U U U > U U U > > R R R B B B L L L F F F > R L R B F B L R L F B F > R R R B B B L L L F F F > > D D D > D D D > D D D > >is solvable in twelve qtw, either by (FFBB)(UD')(LLRR)(UD') or by its >inverse, (DU')(LLRR)(DU')(FFBB). It is immediate that a twelve qtw >path from this pattern to START can begin with a twist of any face in >either direction. The program was used to verify that there are no ten >qtw paths. (It generated the set of positions attainable at most five >qtw from START and the set of positions obtainable from the four-spot >in at most five qtw, and verified that the intersection of the two sets >is empty.) Thus the four-spot is exactly twelve qtw from START and all >its neighbors are exactly eleven qtw from START, proving that the >four-spot is a local maximum. > Call the 4-spot s. Then, the twelve neighbors form two M-conjugacy classes: N1={sL,sL',sF,sF',sR,sR',sB,sB'} and N2={sU,sU',sD,sD'}. Also, we have s'=s. Dan and Jim's solution starts in N1 and ends with a quarter-turn from N2, and since s'=s, we can say "or vice versa". Hence, we can start a solution with any of the twelve quarter turns, and therefore s is a local maximum. There are other positions with the same symmetry characteristics as the 4-spot. That is, there are other positions for which the symmetry group contains sixteen elements. There are only three subgroups of M containing sixteen elements, and the three subgroups are M conjugate. The three M-conjugates of the 4-spot position correspond to the three conjugate subgroups of M containing sixteen elements. But what of other positions with the same symmetry group? For example, if the edges of the 4-spot are all flipped, is the position a local maximum? I don't know, but it is interesting to see how far we can get without knowing a process. Call the 4-spot with all edges flipped t. Then, we certainly have t'=t. Is this true of all positions whose symmetry group contains sixteen elements? Also, we certainly have the twelve neighbors forming M-conjugacy classes similar to those for s, N1 with eight elements and N2 with four. Is this true of all positions whose symmetry group contains sixteen elements? Finally, a solution either starts in N1 or starts in N2. If starting in N1 implies ending with a quarter-turn from N2 or vice versa, then t is a local maximum. Can we prove such a thing without actually finding a solution? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU